CHAPTER 6 THE HYDROGEN ATOM OUTLINE. 3. The HydrogenAtom Wavefunctions (Complex and Real)
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1 CHAPTER 6 THE HYDROGEN ATOM OUTLINE Homework Questions Attached SECT TOPIC 1. The Hydrogen Atom Schrödinger Equation. The Radial Equation (Wavefunctions and Energies) 3. The HydrogenAtom Wavefunctions (Complex and Real) 4. Use of the Wavefunctions (Calculating Averages) 5. The Radial Distribution Function 6. Atomic Units
2 Chapter 6 Homework 1 e Z 1. The energy levels of hydrogenlike atoms are given by: E n 4 0a0 n (a) Calculate the wavelength, in nm, of the n=6 to n=3 radiative transition in He +. (b) Calculate the ionization energy of He +, in ev Note: e 4 0a0 65 kj / mol. The 1s and s wavefunctions of hydrogenlike atoms are: 1s A 1s e A 1 Zr / a0 Zr / a0 s s a 0 (a) Prove the 1s and s are orthogonal to each other. (b) Calculate the normalization constant, A1s. (c) Calculate the most probable value of r for an electron in a 1s orbital. (d) Calculate <r> for an electron in a 1s orbital (Note: first normalize the radial distribution function). (e) Calculate the probability that the electron in a 1s orbitl is between r=0 and r=a0/z (f) Calculate the probability that r is in the range: a0/z r 4a0/Z (g) Calculate the average potential energy of an electron in a =1s orbital. (h) Show that the Radial component of 1s is an eigenfunction of the radial Schrödinger equation (below) with l = 0 Zr e 3. The px wavefunction of hydrogenlike atoms is given by: Zr /a0 px Are sin cos (a) Calculate the most probably value of r for an electron in a px orbital. (b) Calculate <r > for an electron in a px orbital (Note: first normalize the radial distribution function). 4. One of the wavefunctions of hydrogenlike atoms is: /3 0 A Rr ( ) Y (, ) Are Zr a sin e i lm (a) Show that Ylm(,) is an eigenfunction of the L operator (below). What is the eigenvalue?
3 (b) Show that Ylm(,) is an eigenfunction of the Lz operator (below). What is the eigenvalue? (c) Set up the product of 3 integrals in spherical polar coordinates required to calculate <y >. You do NOT have to perform the integrals. 5. Two of the complex hydrogen atom d (l=) wavefunctions are: i ψ n(r,θ, φ) Rn ( r) sin e i ψ n (r,θ, φ) Rn( r) sin e Use the Euler Relations below to show how these can be combined to yield two real forms of the hydrogen atom d wavefunctions. DATA h = 6.63x10-34 J s 1 J = 1 kg m /s ħ = h/ = 1.05x10-34 J s 1 Å = m c = 3.00x10 8 m/s = 3.00x10 10 cm/s k NA = R NA = 6.0x10 3 mol -1 1 amu = 1.66x10-7 kg k = 1.38x10-3 J/K 1 atm. = 1.013x10 5 Pa R = 8.31 J/mol-K 1 ev = 1.60x10-19 J R = 8.31 Pa-m 3 /mol-k me = 9.11x10-31 kg (electron mass) n! a x xe dx n ax x e dx 0 n x x e e x x dx 1.54 dx 1.97 d R dr l( l 1) Z Radial Schrödinger Equation: R ER m dr r dr r a0r L z Operator: Lˆz i L 1 1 Operaor: L sin sin sin ix ix Euler Relations: e cos( x) i sin( x) and e cos( x) isin( x)
4 Chapter 6 The Hydrogen Atom Slide 1 Outline The Hydrogen Atom Schrödinger Equation The Radial Equation Solutions (Wavefunctions and Energies) The Hydrogen Atom Wavefunctions (Complex and Real) Use of the Wavefunctions (Calculating Averages) The Radial Distribution Function Atomic Units Slide 1
5 The Hydrogen Atom Schrödinger Equation The Potential Energy For two charges, q 1 and q, separated by a distance, r: Force: Potential Energy: If the charges are of opposite sign, the potential energy, V(r), is negative; i.e. attractive. Hydrogenlike Atoms (H, He +, Li +, etc.) -e Ze r Slide 3 The Schrödinger Equation In this equation, m represents the mass of an electron (9.11x10-31 kg). Strictly speaking, one should use the reduced mass,, of an electron and proton. However, because the proton is 1830 times heavier, = m. Therefore, many texts (including ours) just use the electron mass. Because V = V(r), one can solve the equation exactly if the Laplacian is written in spherical polar coordinates, giving: Slide 4
6 ^ L Operator ^ The sole dependence of this equation on or is embodied in the L ^ operator. We learned in Chapter 4 (The Rigid Rotor) that the Spherical Harmonics, Y l,m (, ), are eigenfunctions of L ^. ^ ^ They are also eigenfunctions of L z : Slide 5 ^ One can remove the dependence of this equation on and, embodied in L ^, by assuming that: This gives: ^ Because ^ Slide 6 3
7 We can now remove the dependence on and completely. We now have the Radial Equation for the hydrogen atom. This equation must be solved subject to the boundary condition: R(r) 0 as r. The solution to this equation is non-trivial to say the least. We will just present the solutions below. Note: In retrospect, it should not be surprising that the angular solutions of the hydrogen atom are the same as the Rigid Rotor (Chapter 4). Neither potential energy function (V=0 for the Rig. Rot.) depends on or, and they must satisfy the same angular Boundary Conditions. Slide 7 Outline The Hydrogen Atom Schrödinger Equation The Radial Equation Solutions (Wavefunctions and Energies) The Hydrogen Atom Wavefunctions (Complex and Real) Use of the Wavefunctions (Calculating Averages) The Radial Distribution Function Atomic Units Slide 8 4
8 The Radial Equation Solutions The Third Quantum Number When the equation is solved and the boundary condition, R(r) 0 as r, is applied, one gets a new quantum number, n, with the restriction that: or, equivalently Together with the two quantum numbers that came from solution to the angular equations, one has three quantum numbers with the allowed values: Slide 9 The Radial Wavefunctions The functions which are solutions of the Radial Equation are dependent upon both n and l and are of the form: where = 0.59 Å Bohr Radius Several of the Radial functions are: Slide 10 5
9 The Energies The energy eigenvalues are dependent upon n only and are given by: This expression for the energy levels of hydrogenlike atoms is identical to the Bohr Theory expression. 0 However, the picture of electron motion furnished by Quantum Mechanics is completely different from that of the semi-classical Bohr model of the atom k/16 -k/9 -k/4 Energy 1 -k Slide 11 Show that R 10 (r) is an eigenfunction of the Radial equation and that the eigenvalue is given by E 1 (below). or or Note: the alternative forms above have been obtained using: Slide 1 6
10 Slide 13 Therefore: Slide 14 7
11 Outline The Hydrogen Atom Schrödinger Equation The Radial Equation Solutions (Wavefunctions and Energies) The Hydrogen Atom Wavefunctions (Complex and Real) Use of the Wavefunctions (Calculating Averages) The Radial Distribution Function Atomic Units Slide 15 The Hydrogen Atom Wavefunctions The Complete Wavefunction (Complex Form) Note that these wavefuntions are complex functions because of the term, e im. This does not create a problem because the probability of finding the electron in the volume element, dv = r sin()drdd is given by: Slide 16 8
12 Real Form of the Wavefunctions It is common to take the appropriate linear combinations of the complex wavefunctions to obtain real wavefunctions. This is legal because the energy eigenvalues depend only on n and are independent of l and m; i.e. wavefunctions with different values of l and m are degenerate. Therefore, any linear combination of wavefunctions with the same value of n is also an eigenfunction of the Schrödinger Equation. p wavefunctions (l = 1) Already real Slide 17 p wavefunctions (l = 1) (Cont d) Real Real Slide 18 9
13 p wavefunctions (l = 1) (Cont d) Spherical Polar Coords. x = rsin()cos() y = rsin()sin() z = rcos() Slide 19 d wavefunctions (l = ) Already real Slide 0 10
14 d wavefunctions (l = ) (Cont d.) By the same procedures used above for the p wavefunctions, one finds: Slide 1 Plotting the Angular Functions Below are the familiar polar plots of the angular parts of the hydrogen atom wavefunctions. z x y Slide 11
15 Plotting the Radial Functions R 1s R s R 3s R 1s R s R 3s r/a o r/a o R 1s has no nodes R s has 1 node R 3s has nodes Slide 3 R p R 3p R 3d R p R 3p R 3d r/a o r/a o R p has no nodes R 3p has 1 node R 3d has no nodes In General: (a) A wavefunction has a total of n-1 nodes (b) There are l angular nodes (e.g. s-0, p-1, d-)c (c) The remainder (n-1-l) are radial nodes. Slide 4 1
16 Outline The Hydrogen Atom Schrödinger Equation The Radial Equation Solutions (Wavefunctions and Energies) The Hydrogen Atom Wavefunctions (Complex and Real) Use of the Wavefunctions (Calculating Averages) The Radial Distribution Function Atomic Units Slide 5 Use of Hydrogen-like Atom Wavefunctions To illustrate how the hydrogen-like atom wavefunctions may be used to compute electronic properties, we will use the p z (=p 0 ) wavefunction. Review: Spherical Polar Coordinates r 0 r < Distance of point from origin (OP) 0 Angle of vector (OP) from z-axis 0 Angle of x-y projection (OQ) from x-axis Slide 6 13
17 Wavefunction Normalization Slide 7 Slide 8 14
18 We could have combined the extra Z/a 0 into the normalization constant Slide 9 Calculation of <r> and Same as before Slide 30 15
19 Slide 31 Calculation of other Averages We use the same procedures. For example, I ll set up the calculation for the calculation of <y >, where y = rsin()sin() and evaluate. Slide 3 16
20 Outline The Hydrogen Atom Schrödinger Equation The Radial Equation Solutions (Wavefunctions and Energies) The Hydrogen Atom Wavefunctions (Complex and Real) Use of the Wavefunctions (Calculating Averages) The Radial Distribution Function Atomic Units Slide 33 The Radial Distribution Function Often, one is interested primarily in properties involving only r, the distance of the electron from the nucleus. In these cases, it is simpler to integrate over the angles, and. One can then work with a one dimensional function (of r only), called the Radial Distribution Function, which represents the probability of finding the electron between r and r+dr. Slide 34 17
21 The Radial Distribution Function R p R 3p R 3d r/a o Is the most probable value of r for an electron in a hydrogen p orbital the maximum in R p? No!! Relative values of R p represent the relative probability of finding an electron at this value of r for a specific pair of values of and. To obtain the relative probability of finding an electron at a given value of r and any angle, one must integrate * over all values of and. Slide 35 One can write the wavefunction as: The probability of finding an electron at a radius r, independent of and is: or where i.e. we ve incorporated the integrals over and into B is called the Radial Distribution Function (or Radial Probability Density in this text) Slide 36 18
22 We ll use RDF(r) P(r) RDF 1s RDF s RDF 3s r/a 0 RDF 1s has 1 maximum RDF s has maxima RDF 3s has 3 maxima This is because: R 1s has no nodes R s has 1 node R 3s has nodes Slide 37 We ll use RDF(r) P(r) RDF 1s RDF p RDF 3d r/a 0 RDF 1s has 1 maximum RDF p has 1 maximum RDF 3d has 1 maximum This is because: R 1s has no nodes R p has no nodes R 3d has no nodes Slide 38 19
23 Most Probable Value of r RDF 1s RDF p RDF 3d r/a 0 The most probable value of the distance from the nucleus, r, is given by the maximum in the Radial Distribution Function, P(r) RDF(r) It can be computed easily by: Slide 39 The wavefunction for an electron in a p z orbital of a hydrogenlike atom is: We will determine the most probable distance of the electron from the nucleus, r mp. Slide 40 0
24 Therefore: One gets the same result for any p orbital because the Radial portion of the wavefunction does not depend on m. Slide 41 RDF 1s RDF p RDF 3d r/a 0 By the same method, one may calculate r mp for 1s and 3d electrons: These most probable distances correspond to predicted radii for the Bohr orbits: Slide 4 1
25 Probability of r in a certain range We will again consider an electron in a p orbital, for which: Some Numerical Integrals What is the probability that 0 r 5a 0 /Z We will first normalize the RDF Slide 43 Some Numerical Integrals What is the probability that 0 r 5a 0 /Z Define: Then: Slide 44
26 Some Numerical Integrals What is the probability that 3a 0 /Z r < One can use the identical method used above to determine that: Slide 45 Calculate <r> for an electron in a p z orbital (same as worked earlier using the complete wavefunction) Same result as before. Slide 46 3
27 Calculate the average potential energy for an electron in a p z orbital. Slide 47 for an electron in a p orbital Total Energy: Note that <V> = E Calculation of average kinetic energy, <T> Signs <V> negative <T> positive <E> negative Also: Slide 48 4
28 Outline The Hydrogen Atom Schrödinger Equation The Radial Equation Solutions (Wavefunctions and Energies) The Hydrogen Atom Wavefunctions (Complex and Real) Use of the Wavefunctions (Calculating Averages) The Radial Distribution Function Atomic Units Slide 49 Atomic Units All of these funny symbols flying around are giving me a headache. Let s get rid of some of them. Let s redefine some basic units: m e =1 (mass of electron) e = 1 (charge of electron) ħ = 1 (angular momentum) 4 o = 1 (dielectric permittivity) Derived Units Length: Conversions 1 bohr = 0.59 Å Energy: 1 h = 65 kj/mol 1 h = 7.1 ev Slide 50 5
29 Hydrogen Atom Schrödinger Equation SI Units Atomic Units Slide 51 6
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